80 
Proceedings of the Eoyal Society of Edinburgh. [Sess. 
and we define the function 
M kt v {x, y) = L ^ 2 (/x + v-k+l; + 1, 2r + 1 ; x, y). 
The development in ascending powers of x and y is then 
,, V+ 1 _*±e y y (fi + v-k+l, m + n) x m y v 
2 V 2 ““(2/*+ 1, m)(2v+ 1, n) m\ n\ ' 
We see that this function exists only when /u and v are both different 
from the half of a negative integer ; a similar feature occurs with the 
one-variable confluent hypergeometric function 
M*, „(*) = xT H e^ lim F (ix-k + i, P ; 2^+1 ; - ) 
which disappears if 2/ul is a negative integer. If, however, we suppose v, 
for instance, to become equal to — J, and simultaneously y to become equal 
to zero, with the condition that the fraction 0 ^ — tends to zero, the 
l i/+.l 
function becomes, as it is easy to verify by considering the above 
expansion, equal to 
+ k m ) ^ 
" » (v+i , m ) m ! 
or precisely M fc> ^(x). We then have the most important relation 
Mfc, 0) = Mfc, M (z), 
provided that lim ^ =0 ; to which can be added the similar one 
Mfc,_£ >v ( 0 , y) = M ktV (y), 
. OC 
provided that lim — - = 0. 
Lfx -f- 1 
It is easy to form the system of partial differential equations satisfied 
by Mfc f v : it is 
x 2 r - xyq + z(^ - ~ ^ -f kx + \ - /x 2 ^ = 0 
yH - xyp + z(^ - 1 ^ + ky + \ - v 2 ^ = 0. 
(S) 
If in this system we take y = 0 and v=— J, the second equation 
vanishes, and the first one becomes 
o cPz 
dx i 
+ kx + \ - /x 2 ) = 0, 
which is precisely the confluent hypergeometric equation of one variable, 
in Whittaker’s form; and we obtain a similar result by taking x = 0 and 
U— ~~ 2* 
